Abstract

We will first prove the existence of - and -absorbing sets for the three-dimensional primitive equations of large-scale atmosphere in log-pressure coordinate and then prove the existence of -absorbing set by the use of the elliptic regularity theory. Finally, we obtain the existence of - and -global
attractors for the three-dimensional viscous primitive equations of large-scale atmosphere in
log-pressure coordinate by using the Sobolev compactness embedding theory.

1. Introduction

The fundamental equations governing the motion of the atmosphere consist of the Navier-Stokes equations with Coriolis force and thermodynamics, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. Moreover, due to the shallowness of the atmosphere, that is, the depth of the fluid layer is very small in comparison with the radius of the earth, the vertical large-scale motion in the atmosphere is much smaller than the horizontal one, which in turn leads to modeling the vertical motion by hydrostatic balance ([1–3]). But for large-scale atmospheric flows, it is often necessary to take account of compressibility. This may be done most easily by a change of vertical coordinates to log-pressure coordinate rather than a function of geometric height. As a result, one can obtain the system (1)–(4), which is known as the primitive equations for large-scale atmosphere in log-pressure coordinate.

In recent years, the primitive equations of the atmosphere, the ocean, and the coupled atmosphere-ocean have been extensively studied from the mathematical point of view ([1, 4–13]). The mathematical framework of the primitive equations of the ocean was formulated, and the existence of weak solutions was proved by Lions et al. in [11]. In [1], the authors have studied the primitive equations of large-scale atmosphere in pressure coordinate, where the continuity equation is changed into the incompressible equation, but the thermodynamical equation is very complicated. For the sake of simplicity, the authors have made some approximation of the thermodynamical equation, and in this case, the authors have proved the existence and the uniqueness of strong, local-in-time solutions. By assuming that the initial data is small enough, the authors have studied the existence of strong, global-in-time solutions to the three-dimensional primitive equations of large-scale ocean and obtained the existence of strong, local-in-time solutions to the equations for all initial data in [14]. In [5], the authors have proved the maximum principles of the temperature for the primitive equations of the atmosphere in pressure coordinate. The existence and the uniqueness of strong, global-in-time solutions to the primitive equations in thin domains for a large set of initial datum whose sizes depend inversely on the thickness were established in [9]. In [15], Temam and Ziane have considered the existence of strong, local-in-time solutions for the primitive equations of the atmosphere, the ocean, and the coupled atmosphere-ocean. Asymptotic analysis of the primitive equations under the small depth assumption was established in [10]. In [8], the authors have proved the existence of weak solutions and a trajectory attractors for the moist atmospheric equations in geophysics. The existence and the uniqueness of strong, global-in-time solutions for the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics were established by the authors in [4, 16]. In [6], the authors have considered the long-time dynamics of the primitive equations of large-scale atmosphere and obtained a weakly compact global attractor which captures all the trajectories with respect to the weak topology of . The existence and the uniqueness of strong solutions, the existence of smooth solutions, and the existence of a compact global attractor in for the primitive equations with more regular initial data than that in [6] were established in [7]. In [12], the authors have proved the existence and the uniqueness of -weak, global-in-time solutions when the initial conditions satisfy some regularity. The existence of global attractor in for the primitive equations with initial datum in and the heat source was proved by use of the Aubin-Lions compactness lemma in [17]. In [18], the authors have proved the existence of the global attractor in for the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics. In this paper, we will consider the primitive equations of large-scale atmosphere in log-pressure coordinate, where the continuity equation is transformed into the incompressible equation with the weighted function , which is more complicated than that in pressure coordinate, but we do not need to make any approximation of the thermodynamical equation such that the accuracy of the temperature is higher than that in pressure coordinate which will reveal the real world better. Moreover, the appearance of the weighted function makes the calculations more difficult. In order to make sure of the existence of the absorbing set, we need to add some additional conditions (10) on , which means that the thickness of is very small. In [18], there are not any restrictions on the thickness of to ensure the existence of the absorbing sets, and the estimate of can be obtained directly by taking the inner product (15) with . But in this paper, thanks to the appearance of the weighted function , we have to first estimate in and then in to obtain a priori estimates on in . In this paper, we will prove the existence of -global attractor for the three-dimensional primitive equations of large-scale atmosphere in log-pressure coordinate under the assumptions (10) on .

In this paper, we will consider the following three-dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinate [3]:
in the domain
where is a bounded domain in with smooth boundary. Here, , is the velocity field, is the temperature, is the geopotential, is the Coriolis parameter, is the vertical unit vector, is a positive constant, and is the heat source. The operators and are given by
where and are positive constants representing the horizontal and the vertical Reynolds numbers, respectively, and and are positive constants which stand for the horizontal and the vertical heat diffusivities, respectively. And is used as a vertical coordinate, where is a constant “scale height” and is a constant reference pressure. For the sake of simplicity, let be the horizontal gradient operator and the horizontal Laplacian.

We denote the different parts of the boundary by

We equip (1)–(4) with the following boundary conditions, with nonslip and nonflux on the side walls and bottom [4]:
where is a positive constant.

In addition, (1)–(8) are supplemented with the following initial conditions:
Additionally, assume that satisfy

This paper is organized as follows. Section 2 is devoted to introduce the mathematical framework and the definition of solutions for (1)–(10) and give some auxiliary lemmas used in the sequel. In Section 3, we will give the existence and the uniqueness of solutions for (1)–(10). Next, some a priori estimates of solutions corresponding to (1)–(10) will be established, which imply the existence of - and -absorbing sets for (1)–(10). In Section 4, we first obtain the existence of the -global attractor for (1)–(10) by compactly, and then get the existence of -absorbing set of (1)–(10) by the elliptic regularity theory. Finally, we prove the existence of the -global attractor for the three-dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinate by the Sobolev compactness embedding theory.

Throughout this paper, let be positive constants independent of which may be different from line to line.

Integrating the continuity equation (3) and taking the boundary conditions for into account, we obtain

And then, we consider the vertical momentum equation, that is, the second equation (2). First, we define an unknown function on (i.e., ), where is a constant reference pressure, say
which is the geopotential of the atmosphere on .

Then,
Therefore, (1)–(4) can be rewritten as follows:
with the following boundary conditions:
and the initial data
At last, we will divide (14) into two systems with respect to and [4], where and are defined by
where
and we have .

Therefore, satisfies the following systems:
with the boundary conditions
and satisfies the following systems

with the boundary conditions

2.2. Some Function Spaces

In this subsection, we will quote briefly some notations and function spaces used in this paper.

First of all, we introduce the notations for some function spaces on as follows:
Let (or ) be the weighted Hilbert spaces of (or ) functions or (or )-vector valued functions on and denote the usual -norms by , where (or ). Denote the inner product in by and the norm in by , respectively, given by
for any and .

Similarly, define
for any and , respectively.

Let be the closure of with respect to the following norms:
for any and , respectively, and let .

We start with the following general existence and uniqueness of solutions which can be obtained by the standard Fatou-Galerkin methods [20–22] and similar methods in [4]. Here, we only state it as follows.

Theorem 6. Assume that and and let be any fixed positive time. Then, there exists a unique strong solution of (14)–(18) on , which satisfies
and depends continuously on the initial data ; that is, the mapping is continuous in .By Theorem 6, we can define the operator semigroup in as
which is -continuous.

In this section, we will give some a priori estimates which imply the existence of the -absorbing set for (1)–(10). That is, for any bounded subset , there exists such that for any , where is a semigroup on Banach space generated by (1)–(10).

3.1. Estimates of

Taking the inner product of (15) with , using the boundary conditions (17) and using Lemma 3, we deduce that
Employing Hölder’s inequality yields
Thanks to
we obtain
Setting and , we find that
which implies that
Therefore, there exist three positive constants such that
for any . For brevity, we omit writing out explicitly these bounds here, and we also omit writing out other similar bounds in our future discussion for all other uniform a priori estimates.

3.2. Estimates of

Multiplying (14) by , integrating over , and using Lemma 3, we deduce that
Setting and , we obtain
From the classical Gronwall inequality, we have
for all , which implies that there exist three positive constants ,, such that
for all .

3.3. Estimates of

Taking the inner product of (15) with and using Lemma 3, we have
From the inequality , we get
which implies that
Therefore, by virtue of the uniform Gronwall lemma, from (44) and , we deduce that
for any .

3.4. Estimates of

Multiplying (23) by , integrating over , and using Lemma 3, we deduce that
It follows from the Sobolev interpolation inequality and Lemma 4 that
which implies that
where we use the inequality .

We deduce from (54) and (56) that
Therefore, it follows from (44)–(49) and the uniform Gronwall lemma that
for any .

3.5. Estimates of

Taking the inner product of (21) with and combining the boundary conditions (22), we deduce that
By using the Hölder inequality, we get
It follows from (60) and (61) that

In view of (44)–(49) and (59) and the uniform Gronwall lemma, we obtain
for any .

3.6. Estimates of

Denote that . It is clear that satisfies the following equation obtained by differentiating (14) with respect to :
with the boundary conditions
Multiplying (64) by , integrating over , and using Lemma 3, we get
It follows from the Young inequality and the Sobolev embedding theorem that

Since
which implies that
for any .

By virtue of the uniform Gronwall lemma, from (43), (49), and (69), we deduce that
for any .

3.7. Estimates of

Taking the inner product of (14) with and combining the boundary conditions (16), we deduce that
We derive from (72) and the Young inequality that

By virtue of (44)–(49), (69)–(71), and the uniform Gronwall lemma, we get
for any .

3.8. Estimates of

Denote . It is clear that satisfies the following equation obtained by differentiating (15) with respect to :
with the boundary conditions
Multiplying (76) by , integrating over , and using Lemma 3, we deduce that
We deduce from the Young inequality that
It follows from (78) and (79) that

3.11. Estimates of

Taking the inner product of (76) with , combining the boundary conditions (77), and using Lemma 3, we deduce that

It follows from the Hölder inequality that
We derive from the Young inequality that

As a consequence of the uniform Gronwall lemma, we get
for any .

3.12. Estimates of

Denote that , . It is clear that satisfy the following equations obtained by differentiating (14) and (15) with respect to , respectively,
with the boundary conditions
Multiplying (94) and (95), respectively, by , , integrating over , and using Lemma 3, we deduce that
It follows from the Young inequality that
Multiplying (15) by and integrating over , we obtain
Similarly, we have
We derive from the Young inequality that